Further, (REF ) and (REF ) imply \(2\sigma _k + (2\chi - \varphi )/\tilde{\varphi }>0\) .
Thus, Lemma  leads to \(\langle \mathcal {J}_{4,k}\rangle _t = O(\chi _t^2/\varphi _t)\) ; and the strong law of large number [1]} suggests that
\(\mathcal {J}_{4,t,k} = o{\sqrt{\chi _t^2/\varphi _t \cdot \lbrace \log (\varphi _t/\chi _t^2)\rbrace ^{1+\upsilon ^{\prime }} }} = o{\sqrt{\chi _t^2/\varphi _t\cdot \lbrace \log (1/\chi _t)\rbrace ^{1+\upsilon ^{\prime }} }} \stackrel{(\ref {cond:chi})}{=} o (\chi _t/\varphi _t).\) 
